Inconsistent nonstandard arithmetic

Abstract

This paper continues the investigation of inconsistent arithmetical structures. In §2 the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In §3 several nonisomorphic inconsistent models with identity which extend the (=, <) theory of the usual classical denumerable nonstandard model of arithmetic are exhibited. In §4 inconsistent nonstandard models of the classical theory of finite rings and fields modulo m, i.e. Z_m, are briefly considered. In §5 two models modulo an infinite nonstandard number are considered. In the first, it is shown how to model inconsistently the arithmetic of the rationals with all names included, a strengthening of earlier results. In the second, all inconsistency is confined to the nonstandard integers, and the effects on Fermat's Last Theorem are considered. It is concluded that the prospects for a good inconsistent theory of fields may be limited.